Question

Compute the derivative of the given function. $$f(x)=(1-x)^{3}$$

Short answer

The derivative is \( f'(x) = -3(1-x)^2 \).

Step-by-step solution

Identify the Differentiation Rule Needed

The function given is \( f(x) = (1-x)^3 \). We will use the chain rule to differentiate it, which is applicable when you have a composite function, i.e., one function inside another.

Apply the Chain Rule

The chain rule states that if \(y = u^n\) where \(u\) is a function of \(x\), then \( \frac{dy}{dx} = n \, u^{n-1} \, \frac{du}{dx}\). In this case, set \(u = 1-x\) and \(n = 3\).

Differentiate the Outside Function

Differentiate the outer function \( g(u) = u^3 \) with respect to \(u\). This gives \( 3u^2 \).

Differentiate the Inside Function

Differentiate the inner function \( u = 1-x \) with respect to \(x\), which is \( \frac{du}{dx} = -1 \).

Combine Using Chain Rule

Combine the results from Steps 3 and 4: \( f'(x) = 3(1-x)^2 \cdot (-1) \).

Simplify the Expression

Simplify the expression to get the final derivative: \( f'(x) = -3(1-x)^2 \).

Key concepts

Chain Rule

The chain rule is a crucial differentiation technique used in calculus when dealing with composite functions. It helps us differentiate functions that are composed of two or more simpler functions. Think of it like peeling layers of an onion, focusing on one layer at a time. The chain rule states that if you have a composite function, namely one function inside another, you take the derivative of the outer function first and multiply it by the derivative of the inner function.

Imagine you're working with a function in the form of \( f(g(x)) \). The chain rule provides a straightforward formula: if \( y = g(x) \) then, \[ \frac{dy}{dx} = \frac{dy}{dg} \times \frac{dg}{dx} \].

• First, find the derivative of the outer function, keeping the inner function intact.
• Next, multiply by the derivative of the inner function.

By using the chain rule, you can solve complex derivative problems quite easily! In our exercise, the chain rule made it possible to deal with the composite structure of \( (1-x)^3 \).

Composite Function

Composite functions occur when one function is placed inside another, essentially creating a function of a function. These are everywhere in calculus and require special attention when you want to differentiate them. Let's break down what a composite function really is.

If we have two functions, \( f(x) \) and \( g(x) \), a composite function can be expressed as \( f(g(x)) \). This notation hints that you first apply \( g \) and then apply \( f \). Think of it as a process:

• First, transform \( x \) through \( g(x) \).
• Then, take this result and input it into \( f \).

For the exercise, \( f(x) = (1-x)^3 \), the inner function \( u = 1-x \) sits inside the cubing operation. Recognizing this structure allows one to effectively apply the chain rule and efficiently find derivatives.

Differentiation Rules

Differentiation is the process of finding the derivative, or the rate of change of a function. Various differentiation rules help make this task easier across different types of functions.

In calculus, we often rely on several basic rules:

• Power Rule: If \( f(x) = x^n \), the derivative \( f'(x) = nx^{n-1} \).
• Product Rule: Used when differentiating products of two functions.
• Quotient Rule: Utilized when differentiating a quotient of two functions.
• Chain Rule: As previously discussed, for composite functions.

These rules simplify the process and provide a uniform approach to differentiating various types of functions. In the exercise, the power rule was applied as part of the chain rule when differentiating \( u^3 \), yielding \( 3u^2 \). Understanding these rules is key to working through derivatives successfully.